We intend to continue the work done under Grant GM18770, on the theory of probability estimation, one of the most important applications of which will eventually be to medical diagnosis. The problems are different for continuous and discontinuous distributions though some variables are "mixed" in the medical application. For both classes of problem one can use the method of penalized likelihood which has both a Bayesian and a non-Bayesian interpretation. The Bayesian interpretation enables one to say how much better one putative density curve is than another one as measured by the technical concept of weight of evidence. For continuous distributions we have developed this method in detail for univariate problems where the probability density function is not assumed to belong to a parameterized family. We hope to extend this work to bivariate problems. Also we hope we shall be able to look for "bumps" in density curves with the aid of the graphic data tablet because the analytic methods can require a large amount of computer time. For multidimensional contingency tables the "penalty" we subtract from the log-likelihood is proportional to negative entropy. We hope to apply the method of penalized likelihood also to parametric problems. Related to probability estimation is the Bayes/non-Bayes compromise. It has been applied strikingly to significance tests for multinomial distributions (in a published paper), and we have made some progress also with univariate continuous application (in a doctoral thesis). Recently we have applied this method to contingency tables, including multidimensional contingency tables and have submitted the theory for publication. Much of the theory is now being exemplified by computer runs. We intend to apply the method also to multivariate continuous problems, and hope to find new significance criteria having asymptotic distributions reliable into their extreme tails. We hope to compare our methods with those developed elsewhere. It will be possible to do this well only if we obtain a good full-time programmer.